What is the role of convex optimisation in AI systems?
Is there any role of convex optimization in AI? 


Yes, of course!


  If so, in what algorithms or problem settings or systems?


The problem of finding the parameters of a support vector machine can be formulated as a convex optimization problem. Another example is linear regression.

See also the paper Convex Optimization: Algorithms and Complexity (2014) by Sébastien Bubeck, which also mentions SVM as a typical example.